Periods of Strebel Differentials and Algebraic Curves Defined over the Field of Algebraic Numbers

Abstract

In \cite{MP} we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show in this article that even if all geometric data are defined over $\overline{\mathbb{Q}}$, the Strebel differential can still have a transcendental period. We construct a Strebel differential $q$ on an arbitrary complete nonsingular algebraic curve $C$ defined over $\overline{\mathbb{Q}}$ such that (i) all poles of $q$ are $\overline{\mathbb{Q}}$-rational points of $C$; (ii) the residue of $\sqrt{q}$ at each pole is a positive integer; and (iii) $q$ has a transcendental period.